Strategy for finding trapping region of a discrete Dynamical System (Hénon map)

Question

The Hénon map is defined as $H(x,y) = (f(x,y), g(x,y)$ with $f(x,y) = a - x^2 -by$ and $g(x,y) = x$.

The system is known to have an atractor, I want to find a trapping region for this system. My strategy so far has been:

1. Following the discussion Hénon does in A two-dimensional mapping with strange attractor, I take two points $x_1$ and $x_2$ first point is chosen at will $x_2$ is a point close to the unstable fixed point of the system. Iterate the map on $x_1$ and $x_2$, $n = 10000$ times. Ploting the orbits its clear that both converge very quickly to figure 1, and overlap almost completely. This suggests strongly that the system has an atractor.

2. I approximate the atractor by iterating $n= 100000$ times both points and discarding the first $100$ iterations. Once I have a "precise" idea of how the iterator looks like I "perturbed" the points on the orbit by adding random vectors of modulus $\epsilon = 0.001$.

3. Since the Hénon map is invertible I can compute the preimages of the perturbed points very close to the attractor, and see how they evolve.

$H^{-3} (P)$ where $P$ is the perturbed atracttor">

Figure 1 Attractor in red, $H^{-3} (P)$ where $P$ is the perturbed atracttor

$H^{-4}(P)$ in grey ">

Figure 2 Attractor in red, $H^{-4}(P)$ in grey.

Figure 3. Attractor in red, $H^{-5}(P)$ in grey.

I was hopping that this approach gave me some clear idea of how a trapping region would look like. But it has not given any clear shape so far. The points under the inverse map diverge very quickly suggesting that the Basin of Attraction is very big. One of the stratiegies I'm thinking I could follow from here is to try to enclose the attractor with an easy rectangular shape and plot its image. Another idea would be to compute the $\epsilon-$neighbourhood, with $\epsilon$ to be choosen, of my approximation of the attractor and compute its image.

If in any of the cases the image falls inside the original set I would have found a trapping region.

Is there any other more "standard" approach?

Also even if could find such region and the simulations would give a positive result how would I prove the existence of a trapping region rigorously?

I'm thinking a way to approach this would be to bound the distance of the images any two points $x_1$ $y_1$, $H(x,y) < Ld(x,y)$ where $L$ is a constant and then choose the maximum distance allowed between points in the border of the trapping region $U$ such that $Ld(x,y) < d(H(\partial U), \partial U)$.